Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-06T13:03:27.792Z Has data issue: false hasContentIssue false

PROPERTIES OF THE INVERSE OF A NONCENTRAL WISHART MATRIX

Published online by Cambridge University Press:  27 May 2021

Grant Hillier*
Affiliation:
University of Southampton
Raymond Kan
Affiliation:
University of Toronto
*
Address correspondence to Grant Hillier, Economics Division, School of Social Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK; e-mail: ghh@soton.ac.uk.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.

Type
ARTICLES
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Thanks to the Editor, Co-Editor, and two referees for helpful comments on the first version of the paper.

References

REFERENCES

Andrews, D. W. K. & Stock, J. H. (2005) Inference with weak instruments. In Blundell, R., Newey, W. K., and Persson, T. (eds.), Advances in Economics and Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society, vol. III. Cambridge University Press. 122173.Google Scholar
Bingham, C. (1974) An identity involving partitional generalized binomial coefficients. Journal of Multivariate Analysis 4, 210223.CrossRefGoogle Scholar
Chan, C., Drensky, V., Edelman, A., Kan, R. & Koev, P. (2019) On computing Schur functions and series thereof. Journal of Algebraic Combinatorics 50, 127141.CrossRefGoogle Scholar
Constantine, A. G. (1963) Some non-central distribution problems in multivariate analysis. Annals of Mathematical Statistics 34, 12701285.CrossRefGoogle Scholar
Constantine, A. G. (1966) The distribution of Hotelling’s generalized ${T}_0^2$ . Annals of Mathematical Statistics 37, 215225.CrossRefGoogle Scholar
Davis, A. W. (1979) Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory. Annals of the Institute of Statistical Mathematics 31(Part A), 465485.CrossRefGoogle Scholar
Davis, A. W. (1980) Invariant polynomials with two matrix arguments extending the zonal polynomials. In Krishnaiah, P. R. (ed.), Multivariate Analysis-V. North-Holland Publishing Company.Google Scholar
Davis, A. W. (1981) On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials. Annals of the Institute of Statistical Mathematics 33(Part A), 297313.CrossRefGoogle Scholar
Díaz-García, J. & Gutiérrez-Jáimez, R. (2001) The expected value of zonal polynomials. Test 10, 133145.CrossRefGoogle Scholar
Dumitriu, I., Edelman, A. & Shuman, G. (2007) MOPS: Multivariate orthogonal polynomials (symbolically). Journal of Symbolic Computation 42, 587620.CrossRefGoogle Scholar
Gutiérrez, R., Rodriguez, J. & Saeź, J. (2000) Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials. Electronic Transactions on Numerical Analysis 11, 121130.Google Scholar
Hillier, G. H., Kinal, T. & Srivastava, V. K. (1984) On the moments of ordinary least squares and instrumental variables estimators in a general structural equation. Econometrica 52, 185202.CrossRefGoogle Scholar
James, A. T. (1961) The distribution of non-central means with known covariance. Annals of Mathematical Statistics 32, 874882.CrossRefGoogle Scholar
James, A. T. (1968) Calculation of zonal polynomial coefficients by use of the Laplace–Beltrami operator. Annals of Mathematical Statistics 39, 17111718.CrossRefGoogle Scholar
Jiu, L. & Koutschan, C. (2020) Calculation and properties of zonal polynomials. Mathematics in Computer Science 14, 623640.CrossRefGoogle Scholar
Khatri, C. G. (1966) On certain distribution problems based on positive definite quadratic functions of normal vectors. Annals of Mathematical Statistics 37, 468479.CrossRefGoogle Scholar
Koev, P. & Edelman, A. (2006) An efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of Computation 75, 833846.CrossRefGoogle Scholar
Krishnan, M. (1967) The moments of a doubly non-central t-distribution. Journal of the American Statistical Association 62, 278287.Google Scholar
Kushner, H. B. (1988) The linearization of the product of two zonal polynomials. SIAM Journal of Mathematical Analysis 19, 687717.CrossRefGoogle Scholar
Kushner, H. B., Lebow, A. & Meisner, M. (1981) Eigenfunctions of expected value operators in the Wishart distribution, II. Journal of Multivariate Analysis 11, 418433.CrossRefGoogle Scholar
Kushner, H. B. & Meisner, M. (1984) Formula for zonal polynomials. Journal of Multivariate Analysis 14, 336347.CrossRefGoogle Scholar
Letac, G. & Massam, H. (2008) The non-central Wishart as an exponential family, and its moments. Journal of Multivariate Analysis 99, 13931417.CrossRefGoogle Scholar
Macdonald, I. G. (1995) Symmetric Functions and Hall Polynomials, 2nd Edition. Oxford University Press.Google Scholar
Macdonald, I. G. (2013) Hypergeometric functions, I. Preprint. arXiv:1309.4568v1 [math.CA].Google Scholar
Merca, M. (2015) Augmented monomials in terms of power sums. SpringerPlus 4, 724.CrossRefGoogle ScholarPubMed
Muirhead, R. J. (1982) Aspects of Multivariate Statistical Theory. Wiley.CrossRefGoogle Scholar
Phillips, P. C. B. (1983) Exact small sample theory in the simultaneous equations model. In Griliches, Z. and Intriligator, M. D. (eds.), Handbook of Econometrics, vol. I: North-Holland Publishing Company.Google Scholar
Phillips, P. C. B. (1989) Partially identified econometric models. Econometric Theory 5, 181240.CrossRefGoogle Scholar
Richards, D. S. P. (1982) Differential operators associated with zonal polynomials. I. Annals of the Institute of Statistical Mathematics 34, 111117.CrossRefGoogle Scholar
Stanley, R. (1989) Some combinatorial properties of Jack symmetric functions. Advances in Mathematics 77, 76115.CrossRefGoogle Scholar
Takemura, A. (1984) A Statistical Approach to Zonal Polynomials. Lecture Notes Monograph Series, vol. 4. Institute of Mathematical Statistics.Google Scholar
Ullah, A. (1994) On the inverse moments of the non central Wishart matrix. Parisankhyan Samikkha 1, 4950.Google Scholar